cp's OEIS Frontend

A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

%I A328778 #28 Sep 26 2023 06:29:20
%S A328778 1,3,12,84,588,4116,28812,201684,1411788,9882516,69177612,484243284,
%T A328778 3389702988,23727920916,166095446412,1162668124884,8138676874188,
%U A328778 56970738119316,398795166835212,2791566167846484,19540963174925388
%N A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.
%C A328778 An indecomposable closed walk returns to its starting vertex exactly once (on the final step).
%C A328778 For n > 1, a(n) is the number of 4-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - _Sela Fried_, Sep 25 2023
%H A328778 Colin Barker, <a href="/A328778/b328778.txt">Table of n, a(n) for n = 0..1000</a>
%H A328778 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (7).
%F A328778 G.f.: 2 - 1/f(x) where f(x) is the g.f. for A054879.
%F A328778 From _Colin Barker_, Oct 27 2019: (Start)
%F A328778 G.f.: (1 - 4*x - 9*x^2) / (1 - 7*x).
%F A328778 a(n) = 7*a(n-1) for n>2.
%F A328778 a(n) = 12*7^(n - 2) for n>1.
%F A328778 (End)
%F A328778 E.g.f.: (1/49)*(37 + 12*exp(7*x) + 63*x). - _Stefano Spezia_, Oct 27 2019
%t A328778 nn = 40; list = Range[0, nn]! CoefficientList[Series[ Cosh[x]^3, {x, 0, nn}], x]; a = Sum[list[[i]] x^(i - 1), {i, 1, nn + 1}]; Select[CoefficientList[Series[ 2 - 1/a, {x, 0, nn}], x], # > 0 &]
%o A328778 (PARI) Vec((1 - 4*x - 9*x^2) / (1 - 7*x) + O(x^25)) \\ _Colin Barker_, Oct 28 2019
%Y A328778 Cf. A054879, A025192.
%K A328778 nonn,walk,easy
%O A328778 0,2
%A A328778 _Geoffrey Critzer_, Oct 27 2019